General image de-noising techniques based upon the traditional (orthogonal, maximally-decimated) discrete wavelet-transform (DWT) have proved to provide the state-of-the-art in de-noising performance, in terms of peak signal-to-noise ratio (PSNR), according to many papers presented in the literature, e.g. Crouse M, Nowak R, Baraniuk R (1998) Wavelet-based statistical signal processing using hidden Markov models. IEEE Transactions on Signal Processing 46:886-902, Donoho D (1995) De-noising by soft-thresholding; IEEE Transactions on Information Theory 41:613-627, and Romberg J, Choi H, Baraniuk R (2001) Bayesian tree-structured image modeling using wavelet-domain hidden Markov models; and, IEEE Transactions on Image Processing 10:1056-1068. The basic idea behind this image-de-noising approach is to decompose the noisy image by using a wavelet transform, to shrink or keep (by applying a soft or hard thresholding technique) wavelet coefficients which are significant relative to a specific threshold value or the noise variance and to eliminate or suppress insignificant coefficients, as they are more likely related to the noise. The modified coefficients are then transformed back into the original domain in order to get the denoised image.
Despite the high PSNR values, most of these techniques have their visual performance degraded by the introduction of noticeable artifacts which may limit their use in de-noising of medical images. The common cause of artifacts in the traditional wavelet-based de-noising techniques is due to the pseudo-Gibbs phenomenon which is caused by the lack of translation invariance of the wavelet method. Shift variance results from the use of critical sub-sampling (decimation) in the DWT. Consequently, the wavelet coefficients are highly dependent on their location in the sub-sampling lattice which directly affects the discrimination of large/small wavelet coefficients, likely related to singularities/non-singularities, respectively. Although this problem can be avoided by using an undecimated DWT, it is too computationally expensive.